Oecologia
DOI 10.1007/s00442-014-2945-3
PHYSIOLOGICAL ECOLOGY - ORIGINAL RESEARCH
Deconvolution of isotope signals from bundles of multiple hairs
Christopher H. Remien · Frederick R. Adler ·
Lesley A. Chesson · Luciano O. Valenzuela ·
James R. Ehleringer · Thure E. Cerling
Received: 3 April 2013 / Accepted: 9 April 2014
© Springer-Verlag Berlin Heidelberg 2014
Abstract Segmental analysis of hair has been used in
diverse fields ranging from forensics to ecology to measure the concentration of substances such as drugs and isotopes. Multiple hairs are typically combined into a bundle
for segmental analysis to obtain a high-resolution series of
measurements. Individual hair strands cycle through multiple phases of growth and grow at different rates when in the
growth phase. Variation in growth of hair strands in a bundle
can cause misalignment of substance concentration between
hairs, attenuating the primary body signal. We developed a
mathematical model based on the known physiology of hair
growth to describe the signal averaging caused by bundling
multiple hairs for segmental analysis. The model was used
to form an inverse method to estimate the primary body signal from measurements of a hair bundle. The inverse method
was applied to a previously described stable oxygen isotope
chronology from the hair of a murder victim and provides a
refined interpretation of the data. Aspects of the reconstruction were confirmed when the victim was later identified.
Keywords
methods
Stable isotope · Mathematical model · Inverse
Introduction
Communicated by Hannu J. Ylonen.
Electronic supplementary material The online version of this
article (doi:10.1007/s00442-014-2945-3) contains supplementary
material, which is available to authorized users.
C. H. Remien (*)
National Institute for Mathematical and Biological Synthesis,
University of Tennessee, Knoxville, TN 37996, USA
e-mail: [email protected]
F. R. Adler
Department of Mathematics, University of Utah, Salt Lake City,
UT 84112, USA
F. R. Adler · L. A. Chesson · L. O. Valenzuela · J. R. Ehleringer ·
T. E. Cerling
Department of Biology, University of Utah, Salt Lake City,
UT 84112, USA
e-mail: [email protected]
L. A. Chesson · L. O. Valenzuela · J. R. Ehleringer · T. E. Cerling
IsoForensics, Salt Lake City, UT 84108, USA
T. E. Cerling
Department of Geology, University of Utah, Salt Lake City,
UT 84112, USA
Organic and inorganic substances in mammals are incorporated into hair, remaining inert for relatively long periods of
time (Henderson 1993). Human forensic applications have
long been recognized, as drugs, metabolites, toxins, and
poisons are incorporated into hair at the time of formation
(Cooper et al. 2011; Sachs 1997; Selavka and Rieders 1995;
Staub 1995). More recently, ecologists and anthropologists
have used stable isotopes in hair to infer information about
an animal’s history, including diet, migration, and nutritional
status (Martínez del Rio 2009; Crawford et al. 2008; Wolf
2009; Cerling et al. 2009; Petzke et al. 2010; Schwertl et al.
2003). While the growth of hair is approximately linear,
allowing the measurement of long-term chronology through
segmental sampling, individual hairs typically have too low
linear density to allow high-resolution sampling of a single
hair. Sample sizes vary, but segments for drug testing are
typically between 10 and 30 mm (Cooper et al. 2011) and
generally require 1–300 mg of hair (Wainhaus et al. 1998).
Stable isotope ratios of hydrogen, carbon, nitrogen, oxygen,
and sulfur require ca. 200, 50, 300, 200, and 900 µg, respectively. To achieve these masses for animals with hair of
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Oecologia
relatively low linear density, such as humans, multiple hair
strands are typically aligned at the root and combined into a
single bundle for analysis (Cooper et al. 2011; Lebeau et al.
2011). The challenge is to mathematically extract the true
chemical signal recorded in hair given variations in growth
rate and stasis, because these processes will tend to blur, or
average, the temporal information recorded in hair.
Two distinct processes can cause misalignment between
individual hair strands in a bundle: variation in growth phase
(Sachs 1995) and variation in growth rate (Lebeau et al.
2011). Each hair follicle cycles through three distinct phases
of growth (Chase 1954; Kligman 1959). In humans, approximately 90 % of hairs are typically in the anagen, or growth,
phase, which lasts for 48–72 months. The anagen phase is
followed by a brief catagen phase, during which the follicle
stops producing hair. The hair then enters the telogen, or resting, phase during which the follicle and hair remain dormant
for 2–6 months, until the hair is shed. Variation of growth rate
between individual hairs in the growth phase can also blur
the measured signal of a hair bundle (Lebeau et al. 2011).
The intraindividual coefficient of variation of hair growth in
humans is about 0.1 (Myers and Hamilton 1951; Lee et al.
2005), and similar variation has been observed in horses (West
et al. 2004) and elephants (Wittemyer et al. 2009). Because
individual hair strands vary in growth, signal details present in
a single hair may not be captured in hair bundle measurements
due to averaging misaligned hair strands in a bundle.
Previous studies implicitly assume that the measured bundle signal is equivalent to the signal of a single hair in the
growth phase with average growth rate. Given known physiology of hair growth, we assume that the phase of growth and
growth rates of individual hairs will vary within a bundle. We
have developed a mathematical model of hair growth that
describes the relationship between the signal of a bundle of
hairs and the primary body signal (e.g., the isotopic pool in
the body at equilibrium with hair). The model is based on an
estimation of uncertainty in time since formation, or age, of
the hair at a given length from the root. We used the model to
develop an inverse method to estimate the primary body signal from hair bundle measurements. Our inverse method was
applied to a previously described stable oxygen isotope chronology from a hair bundle of a murder victim (Ehleringer et al.
2010) and provides a refined interpretation of the original data.
φ(t) refers to the input signal to the hair (e.g., the isotopic
pool in the body at equilibrium with hair), not the input signal to the body (e.g., drinking water, drug dosage history,
etc.), and is assumed to be equivalent to the signal of a single hair in the growth phase with the average growth rate.
The expected bundle signal as a function of length is
!
ψ(l) = !(l, τ )φ(τ )dτ ,
(1)
where !(l, a) is the probability density function of time
since formation, or age, of a hair at length l from the root
given that the total length of the hair is at least l and the hair
has not yet been shed. The kernel ! quantifies the lengthdependent uncertainty in age.
Evaluation of !(l, a)
Let !(l, a) be the probability density function for the random variable time since formation, or age, A, of hair at
length l from the root, given that the hair has not been shed,
and the total length of the hair is at least l. We assume that
hair grows at constant random rate R for random time S,
and then rests for random time T before being shed. We
found the distribution of ! in terms of the probability distributions of R, S, and T denoted in Table 1 to be:
! "
l
l
!(l, a) = g(l) 2 h
a
a
#a
+ (1 − g(l))
"
!
l
l
dτ ,
κ(τ
)h
(a − τ )2
a−τ
0
where
g(l) =
!∞
0
h(r)
!∞ !v
s − l/r
fS (s)fT (v − s)dsdvdr
v − l/r
l/r l/r
Table 1 Definitions of distributions used in modeling
Distribution Description
h
H
Probability distribution function for the growth rate of
hair, R
Cumulative distribution function for the growth rate of
hair, R
Methods
fS
Probability distribution function for the time spent in
growth phase, S
Model formulation
FS
Cumulative distribution function for the time spent in
growth phase, S
Mapping single hair signal to hair bundle measurements
fT
Probability distribution function for the time spent in
resting phase, T
Let φ(t) be the time-dependent primary body signal (e.g.,
drug, metabolite, isotope, etc.) to the hair. The function
FT
Cumulative distribution function for the time spent in
resting phase, T
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Oecologia
Table 2 Parameters and distributions used in modeling
Parameter
Description
Value
γ
α
β
ν
Fraction of hairs in resting phase
Coefficient of variation of hair growth rate
Mean hair growth rate
Mean time from beginning of growth phase to hair shedding
0.05, 0.1, 0.3
0.05, 0.1, 0.25
1 cm/month (Lebeau et al. 2011; Myers and Hamilton 1951)
66 months (Sachs 1995)
Distribution
Description
Value
h
Hair growth rate probability density function
fS
Time in growth phase probability density function
fT
Time in resting phase probability density function
"
!
N β, (βα)2
"
!
N ν(1 − γ ), (0.1ν(1 − γ ))2
"
!
N νγ , (0.1νγ )2
and
κ(a) =
!∞
fT (x)
dx.
x
a
Refer to the Electronic Supplemental Material for a
complete derivation.
Averaging due to sample interval
Signal attenuation also occurs by combining material over
the sample length to obtain the mass of material required
for analysis. In the case of sampling a single hair that is
in the growth phase with fixed growth rate r, the expected
measured signal, f (l), is the average of the time-dependent primary body signal, φ(t), converted to length, over the
sample length v:
r
f (l) =
ν
l/r+v/(2r)
!
φ(τ )dτ .
l/r−ν/(2r)
The center of the samples are at lengths l = (2i − 1)ν/2,
where i is a positive integer. Sampling a bundle containing
a large number of hairs allows for small sample length, so
the measured signal of a bundle of hairs approaches ψ(l).
Inverse method
To estimate the time-dependent input signal φ(t) from
measurements of ψ(l), we discretized Eq. 1 into the matrix
vector equation Kf = d, where f is a discretization of φ(t),
d is a vector containing the measured values of ψ(l), and K
is an m × n matrix. The discretization of φ(t) was chosen to
have the same step size as the measurements d.
As in the estimation of a primary input signal
from a measured stable isotope tooth enamel signal described by Passey et al. (2005), we used Tikhonov regularization to invert K, yielding the estimation
!
"−1 !
"
fest = fguess + K T KK T + ε2 I
d − Kfguess , where I is
an identity matrix, fguess is an a priori reference vector, and
ε is a scalar regularization parameter. The reference vector
fguess was chosen to be a vector containing the mean value
of the measurements d.
The damping factor ε was chosen using generalized
cross-validation. The chosen ε minimizes the GCV functional GCV(ε) =
!
1
2
n !rε !
"2
1
n trace(I−Aε )
, where rε = Kfest − d and
!
"−1 T
Aε = K K T K + ε2 I
K (Vogel 2002).
Results
Using the distributions in Table 2, the model depends on
two parameters: γ is the fraction of hairs in the resting
phase (catagen and telogen), and α is the coefficient of
variation of hair growth rate. The integral transform kernel, !(l, a), is the probability density as a function of time
since formation, or age, of hair at a given length from the
root and describes the amount of signal attenuation in the
hair bundle signal. Using the distributions in Table 2, the
kernel !(l, a), with γ = 0.1 and α = 0.1, is shown in Fig. 1.
We varied γ and α to assess how these parameters affect
!(l, a) (Fig. 2). A high fraction of hairs in the resting phase,
γ , results in high variance in age of hair at all lengths and a
mean age that is higher than the length divided by the mean
growth rate. A high coefficient of variation of hair growth
rate, α, results in increasing variance of age with length.
The distribution of age dictates the uncertainty in time
when sampling a single hair and also the expected amount
of averaging from sampling a bundle of multiple hairs.
The assumed stable isotope primary input signals in
Figs. 3 and 4 are from single elephant tail hairs sampled at
5 mm intervals. Elephants have thick hairs with high linear
density, allowing for high-resolution sampling of a single
hair. The data in Fig. 3 represent an individual migrating
between two areas with different δ 15 N values for vegetation
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Oecologia
a
b
1.0
1.5
Fig. 1 The kernel !(l, a) at
lengths 2.5, 5, 10, and 20 (a–d),
respectively, describes the variation of time since formation,
or age, at a given length from
the root
Length = 5 cm
c
d
1.0
1.5
(l, a)
0.0
0.5
Length = 2.5 cm
Length = 20 cm
0.0
0.5
Length = 10 cm
0
5
10
15
20
25
0
5
10
15
20
25
Age (months)
and, thus, an abrupt change in hair δ 15 N occurs at each
migration event (Cerling et al. 2006); transitions between
areas were made in less than 12 h based on GPS-tracking
data. Migration events result in a signal approximating
a step function. Data from this individual are previously
unpublished, but the individual had behavior analogous to
that described by Cerling et al. (2006). We extended this
signal periodically to create a longer signal. The data in
Fig. 4 are from Cerling et al. (2009) and represents gradual
diet transitions coinciding with two rainy seasons per year.
We compared the expected value of the signal from
sampling a bundle of hairs that contains hairs with varying
growth rates and phases, with γ = 0.1 and α = 0.1, to sampling a single hair that has a growth rate of 1 cm/month and
sample length ν = 1.5 cm (Fig. 3). The primary body signal refers to the signal of the body pool at equilibrium with
hair and is the signal of a single hair in the growth phase
with average growth rate. Signal attenuation occurs when
sampling a single hair because of combining material over
the sample length. Combining multiple hairs into a bundle
for sampling attenuates the primary body signal because of
varying growth phases and growth rates of hairs in the bundle. Signal details of sufficiently short duration are highly
damped and are not captured in the bundle signal because
13
of averaging. When sampling a single hair, events of duration shorter than the sample length may not be recorded.
Additionally, when sampling a single hair, there is considerable uncertainty in assigning time to length since the
exact growth rate and phase of growth of the sampled hair
is not usually known.
To test the ability of the inverse method to accurately
reconstruct a primary body signal from measurements,
we used the forward hair bundle model, with γ = 0.1 and
α = 0.1, to create a synthetic measured bundle signal from
an assumed primary body signal. We then used the inverse
method to estimate the primary body signal from the synthetic bundle signal (Fig. 4). The estimated primary body
signal is more similar to the primary body signal (correlation coefficient r 2 = 0.81) than the measured hair bundle
signal is to the primary body signal (correlation coefficient
r 2 = 0.43). High frequency details are lost in the estimated
input signal, but the general pattern remains well estimated.
Application of inverse method to murder victim data
We used Tikhonov regularization to reconstruct the δ 18 O
primary body signal of an unidentified murder victim from
N
8
6
15
4
15
0
5
10 15 20 25 30 35
10
Time (months)
8
N
10 12
Measure
bundle of hairs
4
b
0
5 10
20
30
0
5 10
20
30
Length (cm)
6
Length (cm)
2
4
Fig. 3 Expected length-dependent single hair signal and hair bundle
signal derived from our mathematical model using an assumed timedependent primary body signal. Time indicates months prior to formation of hair at root, length indicates cm from root. The signal of
the measured bundle of hairs is highly attenuated relative to the primary body signal
0
Standard deviation of age (months)
8
6
0
Measure
single hair
15
= 0.1, = 0.1
= 0.3, = 0.25
= 0.3, = 0.03
= 0.05, = 0.25
= 0.05, = 0.03
5
Mean age (months)
Primary body signal
10 12
a
20
25
Oecologia
0
5
10
15
20
25
Length (cm)
Fig. 2 Mean (a), and standard deviation (b), of time since formation,
or age, of hair at a given length from the root for various values of
fraction of hairs in the resting phase, γ , and coefficient of variation of
hair growth rate, α
stable isotope measurements of the organic component of a
hair bundle, with γ = 0.1 and α = 0.1, (Fig. 5a, b) (Ehleringer et al. 2010; Kennedy et al. 2011). The murder victim,
nicknamed by law enforcement “Saltair Sally,” was found
near Salt Lake City, Utah in October 2000. Oxygen stable
isotope measurements of the organic component of hair
can be used to determine the time-dependent geographic
region-of-origin because oxygen isotopes in drinking water
vary with location (Bowen and Wilkinson 2002) and are
incorporated into hair (O’Brien and Wooller 2007; Ehleringer et al. 2008).
The confidence interval for the estimated primary body
signal was obtained by inverting data with noise added to
each measurement, representing uncertainty related to analysis. We created 1,000 replicates of measurements by adding normally distributed noise with mean zero and standard
deviation of 0.15 ‰ to each measurement. For each set of
measurements with added noise, we selected a regularization parameter using generalized cross-validation, and
performed the inversion using Tikhonov regularization to
estimate the primary body signal. We excluded two estimates of the primary body signal because their isotope values lay outside the biologically feasible range of 5–18 ‰.
The shaded region corresponds to δ 18 O within two standard deviations of the mean estimated primary body signal.
We tested the sensitivity of the inversions to changes in γ
and α by increasing and decreasing γ and α by 50 % and
performing the inversions as above. The amplitude of peaks
are most sensitive to α, the coefficient of variation of hair
growth (Figs. S3, S4, S5, and S6). A high value of α leads
to larger amplitude transitions, while a low value of α leads
to inversions that are more damped. The general pattern,
however, is robust to changes in γ and α.
The model inversion yields an estimate of the primary
body pool at equilibrium with the hair, not the input signal to the body. The body equilibrium signal refers to the
primary body signal if it were in instantaneous equilibrium with the input. The body equilibrium signal can be
reconstructed using a previously described method based
on multiple pools with first order kinetics contributing
to the primary body pool that is at equilibrium with hair
(Cerling et al. 2004, 2007). For application to the murder
victim data, we used a single pool with half-life of 7 days,
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Oecologia
C
-23
13
-21
-19
Primary body signal
0
5
10
15
20
25
30
35
30
35
30
35
Time (months)
-21
-23
13
C
-19
Measured hair bundle signal
0
5
10
15
20
25
Length (cm)
-21
-23
13
C
-19
Estimated primary body signal
0
5
10
15
20
25
Time (months)
Fig. 4 A primary body signal that is attenuated from sampling a bundle of hairs can be partially reconstructed from measurements using
inverse methods. Time indicates months prior to formation of hair at
root, length indicates cm from root
States to identify regions with isotope values matching our
predicted range. Regions with matching isotope values
were highlighted in color using ArcGIS 9.3.1®.
The data were originally interpreted as movement
between three isotopically distinct geographic regions,
Regions 1, 2, and 3, corresponding to measured δ 18 O values of approximately 9.9, 8.4, and 9.2 ‰, respectively
(Ehleringer et al. 2010). The transitions from Region
1 to Region 2 and from Region 2 to Region 1 at about
20 and 15 cm, respectively, occur over multiple cm (or
equivalently multiple months) of growth, slow movement
between geographic regions. The transitions from Region
1 to Region 3 and from Region 3 to Region 1 at about 9
and 5 cm, respectively, occur over a short length 10 interval, rapid movement between geographic regions. Region 1
is consistent with the location where the victim was found,
Salt Lake City, UT.
The estimated equilibrium signal differs from the measured hair bundle in several important ways (Figs. 5a, b, S1,
and S2). Transitions between isotopically distinct regions
are more rapid, with more time spent in each region.
Region 2* and Region 3* have lower δ 18 O than Region 2
and Region 3, respectively, and may correspond to different
geographic regions (Fig. S1). Regions 2* and 3* have considerable overlap (Fig. S2), suggesting that they may correspond to the same geographic location. Region 1 is isotopically similar in both the measurements and the estimated
equilibrium signal, and the δ 18 O is consistent with the location where the victim was found. The estimated equilibrium signal has a new short duration region, Region 4, with
δ 18 O of about 11 ‰ that does not appear in the measured
hair bundle signal. Plotted maps were restricted to the western United States. Region 4 also contains a small region in
the northeast United States as well as parts of Canada.
Discussion
which is consistent with observations of hair oxygen stable
isotopes in humans (calculated from O’Brien and Wooller
2007; Ehleringer et al. 2008). In this case, the estimated
equilibrium signal is very similar to the estimated primary
body signal because of rapid turnover of the primary body
pool relative to changes in the input signal (refer to the
Electronic Supplemental Material for more detail).
Region-of-origin maps were created in two steps. For
each region, we used the semi-mechanistic model by Ehleringer et al. (2008) to predict drinking water δ 18 O values
required to produce the hair isotope ratio, with an added
0.5 ‰ to either side of the value. Once the maximum and
minimum δ 18 O values of drinking water were predicted for
each isotopic region, we used the δ 18 O tap water isoscape
produced by Bowen et al. (2007) for the contiguous United
13
Multiple hairs that may be in different growth phases and
have different growth rates are typically combined into a
single bundle of hairs for segmental analysis. We have
developed a mathematical model that describes the signal
averaging caused by bundling hairs for analysis as well as
an inverse method to estimate the primary body signal from
measurements of a hair bundle. If the growth rates of hairs
within a bundle vary substantially or a high proportion of
hairs are in the resting phase of hair growth, the measured
hair bundle signal is highly averaged relative to the primary
body signal, which may lead to misinterpretation of the
signal.
Our model is based on uncertainty in time since formation, or age, of hair a given length from the root. Assigning times to length when sampling a single hair strand also
Oecologia
Region 1
Region 1
8
18
10
Region 1
O
b
Measured hair bundle signal
12
a
Region 2
6
Region 3
0
5
10
15
20
25
Length (cm)
10
6
8
18
O
12
Estimated primary body signal
0
5
10
15
20
25
Time (months)
Region 4
Region 1
Region 1
10
Region 1
8
18
O
12
Estimated equilibrium signal
Region 2*
6
Region 3*
0
5
10
15
20
25
Time (months)
Fig. 5 a Estimated primary body signal and equilibrium signal from hair δ 18 O measurements of a previously described hair bundle of a murder
victim (Ehleringer et al. 2010; Kennedy et al. 2011). b Tap water maps for geographic regions predicted by the estimated equilibrium signal
relies on an estimation of length-dependent age variation.
High variation in growth rates or a high proportion of hairs
in the resting phase of hair growth leads to high uncertainty
in assigning time to measurements along the length of a
single hair strand.
The determination of whether to bundle hairs or sample
a single hair depends on the research question. If precise
timing of an event is critical or high-resolution sampling
is desired to capture events in the recent past, multiple
hairs should be combined into a bundle to decrease sample length and reduce uncertainty in the time corresponding
to measurements. If multiple hairs are to be combined, it
is preferred to combine many hairs rather than few, as the
uncertainty in the bundle signal decreases with the number
of hairs. If the amplitude of a long duration event is more
important or an individual hair strand has high enough
linear density to allow for a high-resolution sample interval, sampling a single hair strand may result in less signal
averaging than sampling a bundle of hairs. When sampling
a single hair, however, there may be high uncertainty in
assigning time to measurements.
The inverse method we developed allows for the estimation of the primary body signal from a measured hair
bundle signal. The method is especially useful in situations
where the primary body signal is averaged, but characteristics of the signal remain. If the averaging is significant
enough, however, noise may dominate, preventing meaningful reconstruction of the primary body signal.
It may be possible to minimize signal averaging from
bundling multiple hairs by ensuring hairs are in the growth
phase. Van Scott et al. have described a technique to examine a hair’s follicle to determine its phase of growth (Van
Scott et al. 1957), though this is only possible if a hair’s
root can be observed (i.e., hair was plucked rather than
cut). Excluding hairs in the resting phase from a bundle has
been shown to reduce the growth cycle error in hair stable
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Oecologia
isotope analysis (Williams et al. 2011). This can be incorporated into our model by reducing γ , the fraction of hairs
in the resting phase, which results in less signal averaging.
If the hair has never been cut, selecting hairs for analysis with small total length also increases the likelihood of
hair being in the growth phase. Since each hair grows for
a period of time before shedding, hairs with small total
length are more likely to be in the growth phase. However,
the bundled signal will still be attenuated due to variation
in growth rate of individual hairs.
We applied our inverse method to a previously described
(Ehleringer et al. 2010; Kennedy et al. 2011) oxygen stable
isotope chronology from the organic component of the hair
of a murder victim. The signal obtained from the inverse
method is an estimate of the δ 18 O of the primary body pool
that is in equilibrium with the hair, rather than the timedependent isotopic composition of drinking water. Applying a model of turnover of the primary body pool δ 18 O to
the estimated primary body signal provides an estimate
of the equilibrium signal. This further sharpens the transitions between regions, though the effect is small due to the
relatively rapid turnover of oxygen isotopes in the primary
body pool at equilibrium with hair. In cases where turnover of the body pools at equilibrium with hair is slow (e.g.,
carbon (Ayliffe et al. 2004) and nitrogen isotopes (Sponheimer et al. 2003)), the signal attenuation caused by turnover of the body pools may be larger than that caused by
bundling multiple hairs. The search maps determined by
the estimated equilibrium signal differ from previously
published interpretations in several important ways. Transitions between isotopically distinct regions are more rapid,
and predicted regions have slightly different δ 18 O. An additional region of short duration with relatively high δ 18 O is
present in the estimated equilibrium, that may have been
averaged by measuring a bundle of multiple hairs.
The murder victim has since been identified through
DNA analysis as Nikole Bakoles. There are several insights
and consistencies between model predictions and observations from Nikole’s life. Nikole was a known traveler,
and the hair record shows this. Nikole is known to have
spent time with her parents, and this shows up in the Seattle-Tacoma predictions. She was a resident of Salt Lake
County, who moved in and out of the area, and this shows
up in the isotope record of her hair. A major unknown is
that the model predicts a third area, north of Salt Lake
City. At the moment, investigators do not know about these
travels.
In some animals, hair grows seasonally. If the growth
phases of individual hair strands are correlated, the signal of a hair bundle will have lower attenuation relative to
the primary body signal because hairs are likely to be in
the same growth phase. There will, however, still be variation in hair growth rate between hairs, and variation in
13
hair growth rate between hairs may have a larger effect on
attenuating the signal (Supplemental Figures S1, S2, S3,
and S4).
Similar methods may also be used to estimate uncertainty
in time since formation, when time cannot be constrained
using other methods, in other length-dependent measurements of ecological tape recorders, such as measurements of
nails, horn, or tropical trees lacking growth rings. Additionally, similar probability density functions may be useful for
estimating the uncertainty in age of an animal or plant from
size or length. The details of these methods will depend on
growth characteristics. Other biological systems, such as
cell-cycle dependent processes, also rely on inverse methods
to estimate individual signal from population-level measurements (Lu et al. 2003; Bar-Joseph et al. 2004; Rowicka et al.
2007; Siegal-Gaskins et al. 2009). In these models, variation
in cell-cycle phase of individual cells can result in important
differences between population measurements and individual cell dynamics.
Hair records the concentration of drugs, toxins, metabolites, and stable isotopes in the body as it grows. Segmental
analysis of a bundle of hair allows for the construction of
a high-resolution record of the history of an animal. However, variation in the growth of individual hair strands can
distort the relation between measurements and the actual
history of the animal. Our mathematical model provides
an estimate of uncertainty in time since formation, or age,
of a hair strand a given length from the root as well as an
inverse method to estimate the time-dependent primary
body signal (e.g., drug, metabolite, isotope, etc.) from segmental analysis of a bundle of hairs, based on measurable
growth characteristics of individual hair strands.
Acknowledgments CHR conducted this work as a University of Utah Research Training Group Fellow through NSF award
#EMSW21-RTG and as a Postdoctoral Fellow at the National Institute for Mathematical and Biological Synthesis, an Institute sponsored by the National Science Foundation, the US Department of
Homeland Security, and the US Department of Agriculture through
NSF Award #EF-0832858, with additional support from The University of Tennessee, Knoxville.
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